$\mathbb C$ isomorphism to $\mathbb{R} \times \mathbb{R}$ under multiplication How can I show, that $(\mathbb C,\cdot)$ is not isomorphic to $(\mathbb{R},\cdot) \times (\mathbb{R},\cdot)$ under multiplication? I tried to point out that $f(1) = 1$, then pair $(1,1) \rightarrow 1 + 0i$, but then I god stuck. I know these semigroups are isomorph under addition, but any reasonable mapping seems not to be even homomorphism (due to fact, that $(a + bi)(c + di) \neq a+c + i(b+d)$. Edit: Sorry, I assumed those were groups.

To show that $(\mathbb{C}, \cdot)$ and $(\mathbb{R}, \cdot)^2$ are not isomorphic as semigroups, observe that both semigroups have a (necessarily unique) annihilator, an element $a$ such that $ax = a$ for any $x$. Both semigroups have a unique element $e$ such that $ex = x$ for any $x \
ot= a$ (in fact the elements other than $a$ form a subsemigroup that is actually a group). The equation $x^2 = e$ has 2 solutions in $(\mathbb{C}, \cdot)$ but 4 solutions in $(\mathbb{R}, \cdot)^2$. Since we can define $a$ and $e$ using only the semigroup operation, the two semigroups cannot be isomorphic.